piano key weir head discharge relationships

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Piano Key Weir Head Discharge RelationshipsRicky M. AndersonUtah State University

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1PIANO KEY WEIR HEAD DISCHARGE RELATIONSHIPS

by

Ricky M. Anderson

A thesis submitted in partial fulfillment

of the requirements for the degree

of

MASTER OF SCIENCE

in

Civil and Environmental Engineering

Approved:

_________________________ _________________________

Blake P. Tullis Michael C. JohnsonMajor Professor Committee Member

_________________________ _________________________Paul J. Barr Byron R. Burnham

Committee Member Dean of Graduate Studies

UTAH STATE UNIVERSITYLogan, Utah

2011


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ii

Copyright Ricky M. Anderson 2011

All Rights Reserved


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iiiABSTRACT

Piano Key Weir Head Discharge Relationships

by

Ricky M. Anderson, Master of Science

Utah State University, 2011

Major Professor: Blake P. Tullis

Department: Civil and Environmental Engineering

A piano key (PK) weir is a type of nonlinear (labyrinth-type) weir developed

specifically for free-surface flow control structures with relatively small spillway

footprints. Currently, no generally accepted standard PK weir design procedure is

available. This is due, in part, to the large number of geometric parameters and a limited

understanding of their effects on discharge efficiency (discharge efficiency is quantified

by the discharge coefficient of the standard weir equation). However, Hydrocoop, a non-

profit French dam spillways association, has recommended a PK weir design and a head-

discharge relationship specific to that geometry.

To develop a better understanding of the effects of PK weir geometry on

discharge efficiency, 13 laboratory-scale, 4-cycle PK and rectangular labyrinth weir

configurations were tested. As a result, the influence of the following PK weir

geometries and/or modifications on discharge efficiency were partially isolated: the inlet-

to-outlet key width ratio, upstream, and downstream apex overhangs; sloped floors;

raising the crest elevation via a parapet wall; fillets underneath the upstream overhangs;


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ivand the crest type. The physical model test matrix also included a PK weir configuration

consistent with the Hydrocoop-recommended design. From the experimental results, the

appropriateness of the Hydrocoop-recommended head-discharge relationship was

evaluated, along with the discharge coefficient behavior associated with the standard weir

equation. Finally, trapezoidal labyrinth weirs were compared to PK weirs to make a

relative comparison of nonlinear weir discharge efficiency; comparisons were made

considering crest length and structure footprint.

(79 pages)


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vACKNOWLEDGMENTS

I would like to thank Blake P. Tullis for providing me the opportunity to be

involved in this research, and for his patience and hours of guidance given while

conducting this research. I would also like to thank my committee members, Michael C.

Johnson and Paul J. Barr, for their support and constructive feedback. Thank you to my

family, friends, and colleagues for their moral support and encouragement. I give special

thanks to my best friend and wife, Marissa B. Anderson, for her enduring patience and

encouragement while I have been involved in this research and beyond.

Ricky M. Anderson


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viCONTENTS

Page

ABSTRACT .................................................................................................................. iii

ACKNOWLEDGMENTS ............................................................................................... v

LIST OF TABLES ........................................................................................................vii

LIST OF FIGURES ....................................................................................................... ix

LIST OF SYMOBLS .................................................................................................... xii

INTRODUCTION ........................................................................................................... 1

LITERATURE REVIEW ................................................................................................ 5

EXPERIEMENTAL SETUP ......................................................................................... 14

TESTING PROCEDURE .............................................................................................. 18

EXPERIMENTAL RESULTS AND DISCUSSION ...................................................... 20

Head-discharge Equations (1) and (2)................................................................. 20Inlet-to-outlet Key Width Ratio (W

i/W

o) ............................................................ 23

Overhangs .......................................................................................................... 25Sloped Floors ..................................................................................................... 27

Fillets ................................................................................................................. 29Parapet Walls ..................................................................................................... 30

Crest Type ......................................................................................................... 32Discharge Efficiency with Multiple Geometric Configurations........................... 33

PK Weirs vs. Trapezoidal Labyrinth Weirs ........................................................ 35

CONCLUSIONS ........................................................................................................... 41

REFERENCES .............................................................................................................. 47

APPENDICES ............................................................................................................... 49

Appendix A: Detailed Drawings of Weirs .......................................................... 50Appendix B: Photographs of Weirs .................................................................... 64


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viiLIST OF TABLES

Table Page

1 Studied PK Weir Geometry .................................................................................. 6

2 Testing Matrix ................................................................................................... 13

3 Trapezoidal Labyrinth Weir Percent Changes in W andLRelative to PKRFHat a ConstantBand QatHt/Pof 0.5................................................................... 38

4 Trapezoidal Labyrinth Weir Percent Changes in Q,B, andLRelative to

PKRFH with Constant WatHt/Pof 0.5 ............................................................. 40


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viiiLIST OF FIGURES

Figure Page

1 Weir parameters on Sharp Crested Linear Weir .................................................... 1

2 Traditional Trapezoidal Labyrinth Weir (A) and PK Weir (B) Geometries ........... 3

3 PK Weir Type-A (A) and Type-B (B) Geometric Parameters ............................... 6

4 PK Weir (PK1.25) ................................................................................................ 12

5 Rectangular Labyrinth Weir (RL) ....................................................................... 13

6 Testing Flume .................................................................................................... 14

7 PK1.25 (A); PK1.25with Fillets, Parapet Wall, and Half-Round Crest(PKRFH) (B) ..................................................................................................... 16

8 Overview of PK Weir Setup in Flume ................................................................ 17

9 Measured and Predicted [per Eq. (2)] Head-Discharge Curves Based onHt(A)

andH(B) ........................................................................................................... 21

10 Cdvs.Ht/PData for 5 Inlet-to-outlet Key Width Ratios (Wi/Wo) ....................... 24

11 PK1.25 atHt/Pof 0.4 ........................................................................................... 25

12 Cdvs.Ht/PData for PK1.25 and RLRIO .............................................................. 26

13 PK1.25 (A) and RLRIO (B) Side Section View at Ht/Pof 0.3 .............................. 26

14 Cdvs.Ht/PData for Rectangular Labyrinth Weirs and PK1.25............................ 28

15 False Sloped Floor Configuration Comparison ................................................... 29

16 Cdvs.Ht/PData for PK1.25 and PKF .................................................................. 30

17 Cdvs.HtData for PK1.25 and PKR ...................................................................... 31

18 Cdvs.HtData for PKRFH and PKRFF .............................................................. 33

19 Cd/Cd (PK1.25) vs.Ht/wData for PK1.25, PKR, PKF, PKRF, and PKRFH............ 34


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xB3 PK Weir with Wi/Wo= 1.25 (PK1.25) with Fillets, Raised Crest, and Half Round

Crest (PKRFH) [Testing was done with and without modifications (fillets, raisedcrest, and half round crest type). Testing PK1.25backwards produced a PK weir

with Wi/Wo= 0.8 (PK0.8)] Photograph ................................................................ 66

B4 Rectangular Labyrinth Weir (RL) Photograph .................................................... 66

B5 Rectangular Labyrinth Weir with Ramps in Inlet and Outlet Cycles (RLRIO)Photograph ......................................................................................................... 67

B6 Rectangular Labyrinth Weir with Ramps in Inlet Cycles (RLRI) Photograph ..... 67

B7 Rectangular Labyrinth Weir with Ramps in Outlet Cycles (RLRO) Photograph . 68


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xiLIST OF SYMBOLS

The following symbols were used in this thesis:

Wi inlet cycle width

Wo outlet cycle width

Bo upstream or outlet cycle cantilever length

Bi downstream or inlet cycle cantilever length

Cd discharge coefficient

g acceleration of gravity

H piezometric head

Ht total head (piezometric head plus velocity head)

L weir length

N weir cycles

n crest length to total weir width ratio (N = L/W)

P weir height

Q discharge

Si slope of inlet cycle or key floor

So slope of outlet cycle or key floor

Ts wall thickness

V velocity

Vup approach velocity

W width of weir

w cycle width


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1INTRODUCTION

With rising demands for increased reservoir water storage, increasing magnitudes

of probable maximum storm events, and the continuing need for dam safety, many

existing spillways are currently undersized and in need of replacement. Reservoir

spillways typically use weirs, gated or non-gated, as the flow control structure. In the

weir head-discharge relationship, Eq. (1), the weirs discharge capacity (Q) is

proportional to the weir length (L).

2

3

23

2

td LHgCQ (1)

In Eq. (1), Qis the discharge, Cdis the discharge coefficient,g is the gravitational

constant,Lis the crest length, andHtis the total upstream head [piezometric head (H)

measured relative to the weir crest plus velocity head (V2/2g)]. HandHtparameters are

shown in Fig. 1.

Fig. 1.Weir Parameters on Sharp Crested Linear Weir

V /2g

Ht

Q

H


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2In general, there are three methods for increasing the discharge efficiency (as

quantified by the discharge coefficient (Cd) of the standard weir equation) of an

uncontrolled weir spillway when limited by a maximum pool elevation: (1) increasing the

width of the spillway, (2) lowering the spillway crest elevation, and/or (3) increaseL

within the existing spillway footprint by replacing the existing linear weir with a non-

linear (labyrinth-type) weir.

IncreasingLof a linear weir, and consequently the discharge channel width, is

often impractical due the dam geometry and/or economic reasons. In addition to likely

being economically unfeasible, lowering the crest elevation (i.e., lowering the entire

spillway structure) decreases the normal pool elevation, reducing the amount of available

water storage. However, the use of non-linear weirs represents a viable and generally

accepted option.

A labyrinth weir, shown in Fig. 2 (A), is a linear weir, which has been oriented in

a zigzag fashion (thus the term non-linear), increasingL, relative to a linear weir, for a

fixed spillway channel width (W). Despite the fact that labyrinth weir coefficient (Cd),

which is geometry [e.g., side wall angle ()] and discharge dependent, are lower that

linear weir Cdvalues, the increase inLcan increase discharge efficiency, relative to a

linear weir, by 3 to 4 times (Tullis et al. 1995). The increase in discharge efficiency

means that less reservoir storage needs to be reserved for flood routing (increased water

storage) without compromising dam safety.


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4A piano key (PK) weir is a recently developed alternative to traditional labyrinth

weir designs that was developed specifically for smaller control structure footprint

applications. As shown in Fig. 2 (B), two main differences of PK weir designs, relative

to traditional trapezoidal labyrinth weir designs are: (1) the PK weir has a simple

rectangular crest layout (in plan view), essentially creating a labyrinth weir with = 0

(rectangular labyrinth weir), and (2) the PK weir geometry has sloped or ramped inlet and

outlet cycle or key floors. Where the available footprint for the control structure is

limited, the sloped floors cantilever the cycles beyond the spillway footprint providing

the PK weir with a longer crest length relative to traditional labyrinth weir designs with

the same footprint.

In an effort to develop a better understanding of the differences in head-discharge

relationships or discharge efficiencies of PK and labyrinth weirs, as well as develop a

better understanding of the influences of the various PK weir geometric parameters on

discharge efficiency, the following study was undertaken.


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5LITERATURE REVIEW

As stated by Lemprire and Ouamane (2003), the PK weir was originally

developed by Blanc of the University of Briska (Algeria), and Lemprire of Hydrocoop

(France), to facilitate and improve the performance of labyrinth-type weirs installed on

smaller spillway footprints. Over 100 PK weir model studies have been completed since

2000 (Lemprire 2009), although data are not available for all studies. Construction of

the first prototype PK weir, Goulours dam in France, was completed in 2006;

construction of the second prototype PK weir, Saint-Marc dam in France, was completed

in 2008 (Laugier 2007, 2009).

Important geometric parameters, shown in Fig. 3, for PK weir design include the

weir height (P), height of crest to center of sloped floor (Pm), crest centerline length (L),

slope of the inlet cycle or key (Si) and outlet cycle or key (So) floors, footprint or spillway

width (W), footprint length (B), upstream or outlet cycle cantilever length (Bo),

downstream or inlet cycle cantilever length (Bi), inlet cycle or key width (Wi), outlet

cycle or key width (Wo), wall thickness (Ts), cycle width (w; where w = Wi+ WoTs),

and number of cycles (N). Important geometric ratios include the weir crest length over

the spillway width (n = L/W), upstream over downstream weir cantilever lengths (Bi/Bo),

inlet over outlet key width (Wi/Wo), and the relative wall thickness (Ts/P). Two basic PK

weir geometries have been studied; Type-A, which features both upstream and

downstream cantilevered cycles (Bi/Bo= 1 typically), and Type-B, which has a longer

cantilevered upstream cycle (regardless of PK weir type,Bi+Bo= constant typically),

and no cantilevered downstream cycle (Bi/Bo= 0).


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6

(A) (B)

Fig. 3.PK Weir Type-A (A) and Type-B (B) Geometric Parameters

Model studies have been performed on various geometries to determine their

effects on discharge efficiency. A partial list of previous PK weir geometries tested is

presented in Table 1.

Table 1.Studied PK Weir Geometry

Reference Type n Wi/Wo Si So Ts/P

Laugier (2009)&Ribeiro et al. (2007)

A 4.94 1.41 2.04:1 2.04:1 0.07

Lemprire (2009) A 5 1.25 1.8:1 1.8:1 NR

Machiels et al. (2009) A 4.15 1 0.849:1 0.849:1 0.0381

Ribeiro et al. (2009) A 4.94-6.66 1.23-1.57 1.67:1-2.70:1 1.72:1-2.04:1 NR

Laugier (2007) A 5 1.43 2.05:1 2.05:1 0.067

Barcouda et al. (2006) A 6 1.2 2:1 2:1 NRBarcouda et al. (2006) B 6 1.2 1:1 2:1 NR

Ouamane & Lemprire (2006) various 4-8.5 0.67-1.49 NR NR various

Hien, et al. (2006) A 4-7 1.5 NR NR NR

Lemprire & Jun (2005) A 6 1.2 2:1 2:1 NR

Lemprire & Ouamane (2003) A 6 1 1.5:1 1.5:1 NR

Lemprire & Ouamane (2003) B 6 1 0.75:1 1.5:1 NR

*NR = not reported

P PBo BoBi

WoWo

Wi Wi

Cross section A-A

A A B

Cross section B-B

B

Pm

W W

BB


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8close to optimum. Hien et al. (2006) studied PK weirs with Wi/Wo= 1.5, but concluded

that Wi/Wo= 1.2 is likely more efficient, although no data were presented to validate that

claim. In a study done by Ouamene and Lemprire (2006), three PK weirs were tested

with varying Wi/Woratios of 0.67, 1.0, and 1.5. They found that by increasing Wi/Wo, an

increase in efficiency results, but gave little explanation as to why this occurs. Ouamene

and Lemprire (2006) claim that Wi/Wo= 1.2 increased the efficiency by 5%, relative to

Wi/Wo= 1, even though data for a PK weir with Wi/Wo= 1.2 were not presented as part

of that study. Later, Lemprire (2009) proposes Wi/Woratio = 1.25 as close to optimal.

The two prototype PK weirs that have been built at the Goulours and Saint-Marc dams

have Wi/Woratios of 1.43 and 1.41, respectively (Laugier 2007, 2009). All studies

reviewed agree that Wi/Wo> 1.0 produces a greater discharge efficiency than Wi/Wo 0.6, and Wi/Wo= 1.5 produces a moderately

higher discharge efficiency than Wi/Wo= 1.25 atHt/P< 0.6. In general, these finding

are consistent with the findings of Ouamene and Lemprire (2006) who reported that by

increasing the Wi/Woratio relative to 1.0, an increase in discharge efficiency results.

According to the results in Fig. 10, Wi/W

oshould be in the range of 1.25 - 1.5 to

maximize discharge efficiency.

The influence of Wi/Woon the discharge efficiency of the PK weir can, in part, be

explained as follows. As the inlet cycle width increases, the overall effect of head loss

associated with flow entering the inlet cycles decreases and the flow area entering the

inlet key increases, increasing the flow carrying capacity of the inlet cycle. In

consequence to increasing the inlet cycle width, the outlet cycle width decreases

(assuming Wi+ Wo= constant). As the outlet cycle width decreases, its ability to collect

all of the flow from the more efficient adjacent inlet cycles and discharge it downstream

without developing localized submergence conditions decreases. Submergence effects in


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24

Fig. 10.Cdvs.Ht/PData for 5 Inlet-to-Outlet Key Width Ratios (Wi/Wo)

the outlet cycles (regions where the flow depth in the outlet cycle exceeds the weir crest

elevation) can reduce the discharge efficiency of the weir. These are some reasons a

balance of Wi/Woexists.

As the discharge over the weir increased, the upstream apex became less efficient

due to the local submergence at the upstream end of the outlet cycle, as shown in Fig. 11.

The downstream apexes, also shown in Fig. 11, did not experience submergence effects

for the discharges tested.


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25

Fig. 11.PK1.25 atHt/Pof 0.4

Overhangs

The effects of the PK weir upstream key overhangs on discharge efficiency were

isolated by comparing Cddata for the PK1.25and RLRIO (rectangular labyrinth with

sloping false floors installed modeling a PK weir with no overhangs). As shown in Fig.

12, PK1.25 is more efficient (higher Cdvalues) than RLRIO. The effect of the PK weir

upstream overhangs on weir discharge efficiency, in part, is likely related to the nature of

the inlet flow contraction and subsequent energy loss associated with flow entering the

inlet cycles. The PK weir overhang geometry increases the inlet flow area and wetter

perimeter, relative to RLRIO, resulting in a reduction of inlet velocities, flow contraction,

and energy loss. Fig. 13 shows a sectional elevation side view of PK1.25and RLRIO atHt

/P= 0.3; the drop in the water surface profile is more pronounced on RLRIO, indicating a

more significant flow contraction and energy loss condition; the PK1.25water surface

profile is nearly horizontal. This, in part, also explains why the PK weir Type-B

Upstream apex

Downstream apex


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26

Fig. 12.Cdvs.Ht/PData for PK1.25and RLRIO

(A) (B)

Fig. 13.PK1.25 (A) and RLRIO (B) Side Section View at Ht/Pof 0.3


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27(larger upstream overhangs) is more discharge efficient than PK weir Type-A (smaller

upstream overhangs) (Lemprire and Ouamane 2003).

The downstream end of the PK weir outlet key has a larger area and wetted

perimeter, relative to the downstream end of the RLRIO outlet cycle or key, resulting in

more efficient outlet cycle flow exit conditions. In part this explains the why the outlet

keys of PK1.25did not fill with water as fast as the RLRIO at similar flow conditions. As

discharge increases, an increase in PK weir discharge efficiency was also likely

influenced by a reduction in local submergence effects in the outlet keys, relative to a

RLRIO. Both upstream and downstream overhangs likely help to increase in discharge

efficiency of the PK1.25, relative to RLRIO.

Sloped Floors

The RL was also tested with various combinations of false, sloping floor installed

in an effort partially isolate the sloping floor effects of the PK weir. As seen in Fig. 14,

RL is less efficient than PK1.25forHt/P > 0.15, and all rectangular labyrinth weir

geometries [RL, RL with ramps in inlet and outlet keys (RLRIO), RL with ramps in inlet

cycles or keys (RLRI), RL with ramps in outlet cycles or keys (RLRO)] performed very

similarly. This suggests that the sloping floor configuration for PK weirs is likely not a

significant factor influencing the weir discharge efficiency, relative to other geometr ic

parameters such as overhangs and the hydraulic shape of the entrance of the PK weir inlet

key.


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28

Fig. 14.Cdvs.Ht/PData for Rectangular Labyrinth Weirs and PK1.25

It was observed while testing the RL with and without sloping floors, that the

outlet cycles or keys of RL and RLRI (no ramps in outlet keys) filled with water at lower

values ofHt/P(0.3 - 0.4), whereas the outlet cycles or keys of RLRIO and RLRO (ramps

in outlet keys) did not fill until higher values ofHt/P(0.6 - 0.7). Ramps in the outlet

cycles or keys produced modest increases in CdforHt/P> 0.25, by helping to evacuate

water out of the outlet cycles or keys (inducing super critical flow out of the outlet keys);

this is evident in the data (Fig. 15), when comparing RL to RLRO. Comparing data from

RLRI and RL (Fig. 15) indicates that sloping floors in the inlet cycles or keys have a

slightly negative effect on weir performance. A combination of sloping floors in the inlet

and outlet cycles or keys results in a decrease in weir


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29

Fig. 15.False Sloped Floor Configuration Comparison

performance atHt/P < 0.6, and an increase in weir performance atHt/P> 0.6, relative to

RL (Fig. 15); it is expected that PK weir sloped floors have a similar effect.

Fillets

The effect of bull-nosed pier-type fillets installed underneath the upstream

overhangs of the PK weir (PKF), as shown in Fig. 7(B), was an increase in weir

discharge efficiency (higher Cdvalues), relative to PK1.25as shown in Fig. 16. The

modest gains in efficiency of PKF, relative to PK1.25, are due to a decrease in inlet energy

loss associated with the improved flow conditions at the inlet cycle entrances.

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

1.03

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Cdother/Cd

RL

Ht/P

RL RLRIO

RLRI RLRO


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30

Fig. 16.Cdvs.Ht/PData for PK1.25 and PKF

Raised Crest

Raised crest effects were tested by installing 1-inch tall vertical parapet walls

installed on top of the PK1.25weir and featured a flat-top crest type (PKR), as shown in

Fig. 7(B). The addition of the parapet walls increasedPby 13.3%. To avoid the shift in

data associate with the variation inPbetween PK1.25and PKR, and to better isolate the

influences of the parapet wall, relative to the PK1.25geometry, a comparison of Cdvs.Ht

(as opposed to Cdvs.Ht/P) data is presented in Fig. 17. As shown in Fig. 17, installing

parapet walls on the crest of a PK weir increased the discharge efficiency considerably,

which supports the findings of Ribeiro et al. (2009).


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31

Fig. 17.Cdvs.HtData for PK1.25and PKR

Parapet walls seem to increase the discharge efficiency of the PK weir as follows.

As a result of the parapet walls, there is an increase in area in the outlet keys, resulting in

an increase in the flow capacity of the outlet key reducing local submergence effects in

the outlet keys (particularly at the apex of the outlet keys, see Fig. 11).

Within the range of 0.12 < Ht< 0.33-ft, it was observed that the crest on the

upstream side of the weir perpendicular to the flow (upstream end of the outlet keys, see

Fig. 11), had a springing nappe [the nappe detached from the upstream edge of the weir

producing a sharp-crested weir-type nappe (Johnson 2000)]; at all otherHtvalues outside

that range (Ht< 0.12-ft andHt> 0.33-ft) the nappe was clinging (there was no air pocket

under the nappe). In part, this may explain the increase in efficiency associated with to

the parapet wall, as shown in Fig. 17, is less within this range. It is also important to


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32realize that by installing a parapet wall,Pis increased, whereas if no parapet wall is

installed and the PK weir is built with the sameP, an increase in weir length (larger

overhangs) results.

The RL is somewhat representative of a PK weir with a tall parapet wall. The Cd

data in Fig. 14 for the PK1.25and RL weirs show that the PK1.25weir is more discharge

efficient than the RL weir. This suggests that, although the parapet wall can increase the

discharge efficiency of a PK weir, a limit on parapet wall height likely exists above

which the discharge efficiency begins to decrease.

Crest Type

The sloped upstream and downstream PK weir floors make building a crest type,

other than a flat-top crest type, more difficult without adding a parapet wall.

Consequently, a half-round crest was machined on the top of a 1-inch tall parapet wall

and attached to the PK1.25(PKRFH). Gains in efficiency were evaluated by comparing

PKRFH and PKRFF (flat-top crest). At low values ofHt/P, the half-round weir crest was

significantly more efficient than the flat-top crest weir crest; asHt/Pincreases, gains in

efficiency decrease gradually, as shown in Fig. 18.

It was observed that the half round crest type allowed the nappe of the upstream

crest horizontal to the flow (upstream crest of outlet keys) to cling [nappe clings to

downstream edge of the crest (Johnson 2000)] for the entire range tested, whereas the flat

top crest type had a leaping nappe (nappe detached from the downstream edge of the weir

crest) within the range of 0.17


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33

Fig. 18.Cdvs.HtData for PKRFH and PKRFF

behavior. In PK weir design cases where a parapet wall is used, rounded crest shapes

will improve the weir discharge efficiency.

Discharge Efficiency with Multiple

Geometric Configurations

When designing a PK weir, it is likely that more than one geometric modification

(e.g., fillets, parapet wall, crest type, etc.) will be used to increase discharge efficiency.

Fig. 19 presents percent differences in efficiency, relative to the PK1.25weir. In Fig. 19,

the Cdratio vs.Ht/w(w= cycle width) was plotted to eliminate shifts in the data

associated withHt/Pcaused by varying weir heights.

0.2

0.3

0.4

0.5

0.6

0.7

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Cd

Ht/P

PKRFH PKRFF


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34

Fig. 19.Cd/Cd (PK1.25) vs.Ht/wData for PK1.25, PKR, PKF, PKRF, and PKRFH

The appropriateness of superimposing the increases in discharge efficiency

associated with each individual PK weir modification (fillets, raised crest, and crest type)

to predict the Cdvalue of a PK weir with multiple weir modifications was investigated by

comparing data from the PK1.25, PKR, PKF, and PKRF. For example, the increase in Cd,

relative to PK1.25, associated with adding a parapet wall and fillets to the PK1.25 geometry

(PKRF) atHt/wof 0.3, was 7.64%. The superposition approach, which summed the

effects of the parapet wall (4.26%) and fillets (2.81%) efficiency increases, predicted an

increase of 7.07% in Cd(-0.57% relative to actual value). The average difference

between the actual values and superposition values for the entire range ofH/wtested was

+ 0.43%. Though superposition is not an exact predictor of change in Cd, the


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35superposition approach appears to provide a reasonable first-order approximation of the

influence of multiple PK weir geometry modifications on Cd, relative to the PK1.25.

PK Weirs vs. Trapezoidal

Labyrinth Weirs

Tullis et al. (1995) showed that Cd, which is representative of the discharge per

unit weir length, decreases as the trapezoidal labyrinth weir sidewall angle () decreases.

A PK weir is similar to a labyrinth weir with = 0. As a relative comparison of non-

linear weir discharge efficiency, the Cdvs.Ht /Pdata for a PK weir (PKRFH) and

trapezoidal labyrinth weirs with varying values, based on trapezoidal labyrinth weir

with quarter-round crest data published by Willmore (2004), are compared in Fig. 20.

As expected, based on the findings of Tullis et al. (1995), the PK weir Cd datacurve is

relatively consistent with the smaller trapezoidal labyrinth weir data. Somewhat of a

surprise, however, is the fact that the PK weir Cddata fall nearly on top of the = 7

curve rather than below it, as might have been expected with = 0. As discussed

previously, PK weir overhangs result in an increase in discharge efficiency, relative to

RLRIO (modeling a PK with no overhangs or vertical walls); this may, in part, explain

why the PK weir performs similarly to = 7 (trapezoidal labyrinth weir with vertical

walls). In general, the discharge efficiency or discharge per unit weir length of a PK weir

will be smaller than most trapezoidal labyrinth weirs (>7for this specific weir

comparison).


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36

Fig. 20.Cdvs.Ht/PData for Trapezoidal Labyrinth Weirs of Varying and PKRFH

The discharge efficiency of a trapezoidal labyrinth weir of varying or a PK weir

is not only a function of the discharge per unit weir length (Cd) but also the amount of

weir length that will fit within a given footprint restriction (i.e., footprint restricted by W

and/orB). In designing a spillway with given footprint restrictions of WandB, if moreL

can fit within the given footprint restrictions, even if the Cdvalues are lower for that

particular weir geometry, an increase in discharge efficiency at a given value ofHtmay

be realized.

Trapezoidal labyrinth weirs with varying (7, 12, 20, and 35) with half-round

crest shapes were compared with the most efficient PK weir (PKRFH) to determine the

corresponding weir lengths and footprint dimensions required to produce the same Q. P

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Cd

Ht/P

=7 =8 =10 =12

=15 =20 =35 PKRFH


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37was common for all weirs (8.75-in) and the labyrinth weir apex and wall thickness

dimensions were determined using the Tullis et al. (1995) design method. Its important

to note that weir Cdand consequently Qvalues vary withHt/P, and that in the weir

design process, the full range of anticipatedHt/Pvalues should be evaluated. However,

for convenience in this study the PK and trapezoidal labyrinth weirs are compared at a

single commonHt/P value (Ht/P=0.5). In calculating the trapezoidal labyrinth weir

lengths required to match the PK weir Q, some labyrinth weirLvalues corresponded with

non-integer cycle numbers; most prototype labyrinth weirs consist of whole cycles or

whole cycles with a half cycle on one end.

For weir layout purposes, the footprint length (B) was restricted to that of the PK

weir; the footprint width (W) was variable to accommodate the requiredL. Fig. 21

presents a plan view of the PK weir geometry and the trapezoidal labyrinth weir

geometries overlaid onto the PK weir footprint (dashed lines). The percent change in W

andLfor the trapezoidal labyrinth weirs, relative to PKRFH, are presented in Table 3.

It is demonstrated by these comparisons that if footprint restrictions of Wexist, the PK

weir, though producing the smallest discharge efficiency per unit length (Cd), relative to

the trapezoidal labyrinth weirs, it produces the highest discharge efficiency per channel

width (W) atHt/P= 0.5. This is due to the considerable increase in weir length

associated with the PK weir geometry, relative to the trapezoidal labyrinth weirs, for a

given channel width. Table 3 also shows that if Wis not restricted in the weir layout,

using a labyrinth weir can significantly reduce the overall weir length, and thus possibly

the cost of the structure (trapezoidal labyrinths have a shorter weir length, and no

overhangs).


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38

PKRFH = 7o = 12

o = 20

o = 35

o

Fig. 21.PKRFH and Trapezoidal Labyrinth Weirs at Constant QandBatHt/Pof 0.5

Table 3.Trapezoidal Labyrinth Weir Percent Changes in W andLRelative to PKRFH ata ConstantBand QatHt/Pof 0.5

Percent change relative to PKRFH

W L

7 44.1% 1.68%

12 30.4% -40.5%

20 30.0% -74.6%

35 37.0% -110.6%

Alternatively, the discharge capacity of PKRFH was also compared with half-

round crest trapezoidal labyrinth weir designs were Wwas restricted butBwas not, as

shown in Fig. 22 (the PK weir footprint is identified with a dashed line). Percent changes

in QandB, relative to a PK weir, atHt/P= 0.5 are presented in Table 4.


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39

Fig. 22.PKRFH and Trapezoidal Labyrinth Weirs with Constant W

= 7o

= 35o

= 12o

PKRFH

= 20o


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40

Table 4.Trapezoidal Labyrinth Weir Percent Changes in Q,B, andLRelative to PKRFH

with Constant WatHt/Pof 0.5

Percent change relative to PKRFH

Q B L7 14.6% 82.0% 16.0%

12 2.6% 70.0% -36.8%

20 -21.1% 51.3% -111.3%

35 -80.8% 15.1% -280.7%

When the footprint dimensionBis non-restricted, the = 7 and 12trapezoidal

labyrinth weir geometries produced higher Qthan the PK weir for a given channel width

atHt/P= 0.5. The larger values (i.e, = 20 and 35), however, produced considerably

less discharge than the PK weir due to the significant decrease in weir length with

increasing . For applications, such as the crest of a thin concrete dam crest, where the

weir footprint is restricted byBand W, the discharge characteristics of the PK weir are

definitely advantageous. For channel applications where the limits on B and/or W may

not be so stringent, trapezoidal labyrinth weirs may prove to be more hydraulically

efficient and more economical to construct (PK weirs have longer crest lengths and are

likely more difficult to construct due to the overhangs).


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41CONCLUSIONS

To develop a better understanding of the effects of PK weir geometry on

discharge efficiency, and to evaluate the hydraulic performance of a recommended PK

weir design found in the literature (Lemprire 2009), laboratory-scale sectional models

of various PK weir geometries were built and tested. Using the test results, the discharge

equation proposed by Lemprire (2009) [Eq. (2)] was evaluated based on its ability to

estimate the head-discharge relationship, and the influence of the specific upstream head

definition (piezometric vs. total head) on the head-discharge relationship estimation.

The influences of variations in specific geometric parameters on discharge

efficiency of PK weir were also evaluated. The effects of the inlet-to-outlet width ratio

(Wi/Wo) were evaluated by testing PK weirs with varying Wi/Wo(Wi/Wo= 1.5, 1.25, 1.0,

0.8, and 0.67) with every other geometric parameter held constant. The effects of the PK

weir cycle apex overhangs and sloping floors were partially isolated by comparing PK

weir discharge efficiency with that of a rectangular labyrinth weir with the same total

crest length, weir height, inlet cycle widths, outlet cycle widths, total weir width, wall

thickness, and crest shape as the PK weir with various configurations of removable

sloped floors (with sloped floors installed, the rectangular labyrinth weir essentially

models a PK weir with no overhangs). Additionally, the following PK weir geometry

modifications were tested: raising the PK weir crest via a parapet wall, installing fillets

underneath the upstream overhangs creating bull nosed piers, and installing half round (as

opposed to a flat top) crest type.


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42The appropriateness of using superposition to account for the changes in

discharge efficiency associated with multiple PK weir geometry modifications was

investigated. Finally, as a relative comparison of non-linear weir discharge efficiency,

trapezoidal labyrinth weirs were compared with a PK weir according to discharge

efficiency per crest length and for given structure footprint restrictions. Based on the

results of this study, the following conclusions are made.

The linear head-discharge relationship proposed by Lemprire (2009) [Eq. (2)] isnot generally applicable to PK weirs, but rather its specifically applicable to the PK

weir geometry specified by Lemprire (2009) (e.g., Wi/Wo= 1.25, etc.). As Wi/Wo

decreases relative to 1.25, the PK weir head-discharge relationship becomes less

linear (parabolic). Consequently, even if the coefficient (4.3) in Eq. (2) were treated

as a variable specific to different PK weir designs, Eq. (2) would still be limited in its

ability to accurately represent the PK weir head-discharge relationship beyond the

PK weir design recommended by Lemprire (2009).

Lemprire (2009) did not specify whether Eq. (2) was developed based onpiezometric or total head. The results of this study found that using piezometric head

produced an estimate discharge average and maximum error of 1.98% and 3.69%,

respectively, relative to the experimental data. Using the total head produced an

average and maximum percent error of 9.13% and 9.80%, respectively, suggesting

that piezometric head is more appropriate for Eq. (2) when approach velocities are

not negligible (e.g., channel-type applications).


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43

The PK weir design recommended by Lemprire (2009) is presented as a linedrawing that does not include dimensional guidance for all geometric parameters

(e.g. crest shape, wall thickness, shape of weir beneath upstream overhangs, etc.).

The optimal range of Wi/Wofor maximizing discharge efficiency is approximately1.251.5. This is due to the balance of inlet cycle width to outlet cycle width with

respect to hydraulic capacity (ability to convey flow). As the inlet cycle width is

increased, a reduction in energy loss as water enters the inlet keys, as well as an

increase in inlet flow area, results in an increase in discharge capacity; but in

consequence of the inlet key width increasing, the outlet key width is decreased

(assuming Wi+Wo = constant) resulting in a increase in local submergence of the

outlet keys (particularly at the outlet key apexes) and a decrease in outlet key

discharge capacity.

PK weir overhangs result in a measurable increase in discharge efficiency, relative toa rectangular labyrinth weir with sloping false floors (modeling a PK weir with no

overhangs). The PK weir upstream overhang geometry increases the inlet flow area

and wetter perimeter resulting in a reduction of inlet velocities, flow contraction, and

energy loss. This may explain, in part, why the PK weir geometry Type-B (larger

upstream overhangs) is reported to have higher discharge efficiency than PK weir

geometry Type-A (smaller upstream overhangs). The PK weir downstream overhang

geometry results in a larger area and wetted perimeter in the outlet keys, relative to a

rectangular labyrinth weir with false sloped floors, resulting in a more discharge

efficient outlet key exit.


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44

PK weir sloped floors did not significantly influence the weir discharge efficiency,relative to the increase in discharge efficiency due to the PK weir overhangs. False

sloped floors in the outlet keys of the rectangular labyrinth weir aid in reducing local

submergence by helping to evacuate water out of the outlet keys (inducing super

critical flow out of the outlet keys) resulting in an increase in discharge efficiency.

Sloped floors in the inlet keys of the rectangular labyrinth weir have a slightly

negative influence on discharge efficiency. A combination of sloped floors in the

inlet and outlet keys results in a decrease in weir performance atHt/P < 0.6, and an

increase in weir performance atHt/P> 0.6, relative to the rectangular labyrinth weir

with no false sloped floors. It is expected that PK weir sloped floors (inherent in the

PK weir design) have a similar effect.

Installing fillets underneath the upstream overhangs of the PK weir creating a morehydraulic shape, results in an increase in discharge efficiency due to a decrease in

inlet head loss associated with the improved flow conditions at the inlet cycle

entrances.

Raising the crest elevation by installing a parapet wall on the crest of the PK weirresults in an increase in discharge efficiency. This likely results from the increase in

area of the outlet keys, allowing more flow to enter and exit the outlet keys causing a

reduction of local submergence.

Improved crest shapes (half-round vs. a flat-top) results in significant gains indischarge efficiency at low heads; as the head is increased, gains in efficiency

decrease gradually. This is likely the result of the increase in clinging nappe

behavior due to the half round crest type.


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45

If superposition is used to add changes in discharge efficiency resulting from PKweir modifications (i.e., raising a crest via a parapet wall, and installing fillets

beneath the upstream overhangs, improving the crest type) a reasonable first-order

approximation of the change in discharge efficiency will result. In this study,

superimposing increases in discharge efficiency as a result of raising the crest, and

adding fillets beneath the upstream overhangs resulted in an average error of 7.11%.

In general trapezoidal labyrinth weirs are more discharge efficient per crest lengththan PK weirs. If footprint restrictions of length (B) and width (W) exist, the PK

weir, though producing the lowest discharge per unit length, relative to typical

trapezoidal labyrinth weirs, produces the highest discharge efficiency. This is due to

the considerable increase in weir length produced with the PK weir geometry,

relative to the trapezoidal labyrinth weirs, with the same footprint. IfBis restricted,

but Wis not, by increasing W, a trapezoidal labyrinth weir can result in an increase in

discharge efficiency, relative to a PK weir. WhenBis not restricted, trapezoidal

labyrinth weir geometries with smaller side wall angles () produced an increase in

discharge efficiency (higher QatHt/P= 0.5), relative to a PK weir with the same W.

The larger values, however, produced considerably less discharge than the PK weir

due to the significant decrease in total weir length with increasing . For

applications, such as the crest of a thin concrete dam crest, where the weir footprint

is restricted by bothBandW, the discharge characteristics of the PK weir are

advantageous. For channel applications where the limits onBand/or Ware not so

stringent, trapezoidal labyrinth weirs may prove to be more hydraulically efficient


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46and more economical to construct (PK weirs have longer crest lengths and are likely

more difficult to construct due to the overhangs).

Additional research is needed to further investigate optimal values of various

design parameters (e.g. crest type, wall thickness, fillets, floor slope, parapet wall etc.)

on discharge efficiency, and to determine the absolute optimum value of Wi/Wo. In

addition to better understanding PK weir geometry and corresponding discharge

efficiency, additional research may also lead to beneficial additions and/or modifications

of the relatively new general PK weirgeometry.


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47REFERENCES

Barcouda, M., Cazaillet, O., Cochet, P., Jones, B. A., Lacroix, S., Laugier, F., Odeyer, C.,

Vingny, J. P., (2006). Cost-Effective Increase in Storage and Safety of Most DamsUsing Fuse gates or P.K. Weirs. Proc. of the 22nd

Congress of ICOLD., Barcelona,Spain.

Hien, T.C., Son, H.T., & Khanh, M.H.T. (2006). Results of some piano keys weir

hydraulic model tests in Vietnam. Proc. of the 22nd

Congress of ICOLD., Barcelona,Spain.

Johnson, M., C. (2000). Discharge coefficient analysis for flat-topped and sharp-crested

weirs. Irrig. Sci., 19(3), 133-137.

Kline, S.J., McClintock F.A., (1953). Describing Uncertainties in single-sampleExperiments. Mech. Engrg., 75(1), 3-8.

Laugier, F. (2007). Design and construction of the first Piano Key Weir spillway at

Goulours dam. Intl. J. Hydropower & Dams,14(5), 94-100.

Laugier, F. (2009). Design and construction of a labyrinth PKW spillway at Saint-Marcdam, France. Intl. J. Hydropower & Dams,15(5), 100-107.

Lemprire, F. (2009). New Labyrinth weirs triple the spillways discharge.

(Feb. 8, 2010).

Lemprire, F., Jun, G. (2005). Low Cost Increase of Dams Storage and FloodMitigation: The Piano Keys weir. Proc. of 19

thCongress of ICID., Beijing, China.

Lemprire, F., Ouamane, A. (2003). The Piano Keys weir: a new cost-effective

solution for spillways. Hydropower & Dams, 10(5), 144-149.

Machiels, O., Erpicum, S., Archambeau, P., Dewals, B.J., & Pirotton, M. (2009). Largescale experimental study of piano key weirs. Proc. of 33

rdof IAHR., Vancouver,

Canada.

Ouamane, A., Lemprire, F., (2006). Design of a new economic shape of weir. Proc.of the International Symposium of Dams in the Societies of the 21 stCentury, Barcelona,

Spain, 463-470.

Ribeiro, M.L., Bieri, M., Boillat, J.L., Schleiss, A.J., Delorme, F., Laugier, F. (2009).Hydraulic capacity improvement of existing spillways Design of Piano Key weirs.

Proc. of 23rd

Congress of ICOLD., Brasilia, Brazil.


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48Ribeiro, M.L., Boillat, J.L., Schleiss, A., Laugier, F., Albalat, C. (2007). Rehabilitation

of St-Marc damExperimental Optimization of a Piano Key Weir. Proc. of 32nd

Congress of IAHR., Vince, Italy.

Tullis, J. P., Amanian, N., and Waldron. D. (1995). Design of Labyrinth Spillways. J.Hydr. Engrg., 121(3), 247-255.

Willmore, C. (2004). Hydraulic characteristics of labyrinth weirs. M.S. report, UtahState Univ., Logan, Utah.


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49

APPENDICES


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50

APPENDIX A

Detailed Drawings of Weirs


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51

Plan View

Section A - A

Fig. A1.PK Weir with Wi/Wo= 1.5 (PK1.5) Detailed Drawing

Q

36.75"

3.28"

4.91"

0.5"

A A

9.69"

7.75"

19.25"


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52

Plan View

Section A - A


Q

36.75"

3.64"

4.55"

0.5"

A A

9.69"

7.75"

19.25"


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53

Plan View

Section A - A

Fig. A3. PK Weir with Wi/Wo= 1.0 (PK1.0) Detailed Drawing

Q

36.75"

4.09"

4.09"

0.5"

A A

9.69"

7.75"

19.25"


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54

Plan View

Section A - A


Q

36.75"

4.55"

3.64"

0.5"

A A

9.69"

7.75"

19.25"


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55

Plan View

Section A - A


Q

36.75"

4.91"

3.28"

0.5"

A A

9.69"

7.75"

19.25"


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56

Plan View

Section A - A

Fig. A6.PK Weir (PK1.25) with Raised Crest (PKR) Detailed Drawing

Q

36.75"

3.64"

4.55"

0.5"

A A

9.69"

7.75"

19.25"1.0"


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57

Plan View

Section A - A

Fig. A7. PK Weir (PK1.25) with Fillets (PKF) Detailed Drawing

Q

36.75"

3.64"

4.55"

0.5"

A A

9.69"

7.75"

19.25"

2.32"


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58

Plan View

Section A - A

Fig. A8.PK Weir with Raised Crest, Fillets, and Flat Top Crest (PKRFF) DetailedDrawing

Q

36.75"

3.64"

4.55"

0.5"

A A

9.69"

7.75"

19.25"

2.32"

1.0"


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59

Plan View

Section A - A

Fig. A9.PK Weir with Raised Crest, Fillets, and Half Round Crest (PKRFH) Detailed

Drawing

Q

36.75"

3.64"

4.55"

0.5"

A A

9.69"

7.75"

19.25"

2.32"

1.0"


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60

Plan View

Section A - A

Fig. A10.Rectangular Labyrinth Weir (RL) Detailed Drawing

Q

36.75"

3.64"

4.55"

0.5"

A A

7.75"

19.25"


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61

Plan View

Section A - A

Fig. A11.Rectangular Labyrinth Weir with Ramps in Outlet Cycles (RLRIO) Detailed

Drawing

Q

36.75"

3.64"

4.55"

0.5"

A A

9.69"

7.75"

19.25"


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62

Plan View

Section A - A

Fig. A12.Rectangular Labyrinth Weir with Ramps in Inlet Cycles (RLRI) DetailedDrawing

Q

36.75"

3.64"

4.55"

0.5"

A A

13.97"

7.75"

19.25"


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63

Plan View

Section A - A

Fig. A13.Rectangular Labyrinth Weir with Ramps in Outlet Cycles (RLRO) DetailedDrawing

Q

36.75"

3.64"

4.55"

0.5"

A A

13.97"

7.75"

19.25"


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64

APPENDIX B

Photographs of Weirs


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65

Fig. B1.PK Weir with Wi/Wo= 1.5 (PK1.5) [PK1.5 was tested backwards producing a PK

weir with Wi/Wo= 0.67 (PK0.67)] Photograph

Fig. B2.PK Weir with Wi/Wo= 1.0 (PK1.0) Photograph

Q

Q


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66

Fig. B3.PK Weir with Wi/Wo= 1.25 (PK1.25) with Fillets, Raised Crest, and Half RoundCrest (PKRFH) [Testing was done with and without modifications (fillets, raised crest,

and half round crest type). Testing PK1.25backwards produced a PK weir with Wi/Wo=0.8 (PK0.8)] Photograph

Fig. B4. Rectangular Labyrinth Weir (RL) Photograph

Q

Q


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67

Fig B5.Rectangular Labyrinth Weir with Ramps in Inlet and Outlet Cycles (RLRIO)Photograph

Fig. B6.Rectangular Labyrinth Weir with Ramps in Inlet Cycles (RLRI) Photograph

Q

Q


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68

Fig. B7.Rectangular Labyrinth Weir with Ramps in Outlet Cycles (RLRO) Photograph

Q

piano key weir head discharge relationships

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